3.7.34 \(\int \frac {1}{x^{5/2} \sqrt {2-b x}} \, dx\) [634]

Optimal. Leaf size=40 \[ -\frac {\sqrt {2-b x}}{3 x^{3/2}}-\frac {b \sqrt {2-b x}}{3 \sqrt {x}} \]

[Out]

-1/3*(-b*x+2)^(1/2)/x^(3/2)-1/3*b*(-b*x+2)^(1/2)/x^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {47, 37} \begin {gather*} -\frac {\sqrt {2-b x}}{3 x^{3/2}}-\frac {b \sqrt {2-b x}}{3 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*Sqrt[2 - b*x]),x]

[Out]

-1/3*Sqrt[2 - b*x]/x^(3/2) - (b*Sqrt[2 - b*x])/(3*Sqrt[x])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps

\begin {align*} \int \frac {1}{x^{5/2} \sqrt {2-b x}} \, dx &=-\frac {\sqrt {2-b x}}{3 x^{3/2}}+\frac {1}{3} b \int \frac {1}{x^{3/2} \sqrt {2-b x}} \, dx\\ &=-\frac {\sqrt {2-b x}}{3 x^{3/2}}-\frac {b \sqrt {2-b x}}{3 \sqrt {x}}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 25, normalized size = 0.62 \begin {gather*} \frac {(-1-b x) \sqrt {2-b x}}{3 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(5/2)*Sqrt[2 - b*x]),x]

[Out]

((-1 - b*x)*Sqrt[2 - b*x])/(3*x^(3/2))

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 3.30, size = 90, normalized size = 2.25 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\sqrt {b} \left (2+b x \left (1-b x\right )\right ) \sqrt {\frac {2-b x}{b x}}}{3 x \left (-2+b x\right )},\frac {1}{\text {Abs}\left [b x\right ]}>\frac {1}{2}\right \}\right \},-\frac {I b^{\frac {3}{2}} \sqrt {1-\frac {2}{b x}}}{3}-\frac {I \sqrt {b} \sqrt {1-\frac {2}{b x}}}{3 x}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/(x^(5/2)*Sqrt[2 - b*x]),x]')

[Out]

Piecewise[{{Sqrt[b] (2 + b x (1 - b x)) Sqrt[(2 - b x) / (b x)] / (3 x (-2 + b x)), 1 / Abs[b x] > 1 / 2}}, -I
b ^ (3 / 2) Sqrt[1 - 2 / (b x)] / 3 - I Sqrt[b] Sqrt[1 - 2 / (b x)] / (3 x)]

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Maple [A]
time = 0.12, size = 29, normalized size = 0.72

method	result	size
gosper	\(-\frac {\left (b x +1\right ) \sqrt {-b x +2}}{3 x^{\frac {3}{2}}}\)	\(19\)
meijerg	\(-\frac {\sqrt {2}\, \left (b x +1\right ) \sqrt {-\frac {b x}{2}+1}}{3 x^{\frac {3}{2}}}\)	\(22\)
default	\(-\frac {\sqrt {-b x +2}}{3 x^{\frac {3}{2}}}-\frac {b \sqrt {-b x +2}}{3 \sqrt {x}}\)	\(29\)
risch	\(\frac {\sqrt {\left (-b x +2\right ) x}\, \left (x^{2} b^{2}-b x -2\right )}{3 x^{\frac {3}{2}} \sqrt {-b x +2}\, \sqrt {-x \left (b x -2\right )}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(5/2)/(-b*x+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*(-b*x+2)^(1/2)/x^(3/2)-1/3*b*(-b*x+2)^(1/2)/x^(1/2)

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Maxima [A]
time = 0.27, size = 28, normalized size = 0.70 \begin {gather*} -\frac {\sqrt {-b x + 2} b}{2 \, \sqrt {x}} - \frac {{\left (-b x + 2\right )}^{\frac {3}{2}}}{6 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(-b*x+2)^(1/2),x, algorithm="maxima")

[Out]

-1/2*sqrt(-b*x + 2)*b/sqrt(x) - 1/6*(-b*x + 2)^(3/2)/x^(3/2)

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Fricas [A]
time = 0.32, size = 18, normalized size = 0.45 \begin {gather*} -\frac {{\left (b x + 1\right )} \sqrt {-b x + 2}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(-b*x+2)^(1/2),x, algorithm="fricas")

[Out]

-1/3*(b*x + 1)*sqrt(-b*x + 2)/x^(3/2)

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Sympy [C] Result contains complex when optimal does not.
time = 1.13, size = 139, normalized size = 3.48 \begin {gather*} \begin {cases} - \frac {b^{\frac {7}{2}} x^{2} \sqrt {-1 + \frac {2}{b x}}}{3 b^{2} x^{2} - 6 b x} + \frac {b^{\frac {5}{2}} x \sqrt {-1 + \frac {2}{b x}}}{3 b^{2} x^{2} - 6 b x} + \frac {2 b^{\frac {3}{2}} \sqrt {-1 + \frac {2}{b x}}}{3 b^{2} x^{2} - 6 b x} & \text {for}\: \frac {1}{\left |{b x}\right |} > \frac {1}{2} \\- \frac {i b^{\frac {3}{2}} \sqrt {1 - \frac {2}{b x}}}{3} - \frac {i \sqrt {b} \sqrt {1 - \frac {2}{b x}}}{3 x} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(5/2)/(-b*x+2)**(1/2),x)

[Out]

Piecewise((-b**(7/2)*x**2*sqrt(-1 + 2/(b*x))/(3*b**2*x**2 - 6*b*x) + b**(5/2)*x*sqrt(-1 + 2/(b*x))/(3*b**2*x**
2 - 6*b*x) + 2*b**(3/2)*sqrt(-1 + 2/(b*x))/(3*b**2*x**2 - 6*b*x), 1/Abs(b*x) > 1/2), (-I*b**(3/2)*sqrt(1 - 2/(
b*x))/3 - I*sqrt(b)*sqrt(1 - 2/(b*x))/(3*x), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (28) = 56\).
time = 0.00, size = 75, normalized size = 1.88 \begin {gather*} \frac {32 \sqrt {-b} b \left (-3 \left (\sqrt {-b x+2}-\sqrt {-b} \sqrt {x}\right )^{2}+2\right )}{2\cdot 6 \left (\left (\sqrt {-b x+2}-\sqrt {-b} \sqrt {x}\right )^{2}-2\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(-b*x+2)^(1/2),x)

[Out]

-8/3*(3*(sqrt(-b)*sqrt(x) - sqrt(-b*x + 2))^2 - 2)*sqrt(-b)*b/((sqrt(-b)*sqrt(x) - sqrt(-b*x + 2))^2 - 2)^3

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Mupad [B]
time = 0.29, size = 19, normalized size = 0.48 \begin {gather*} -\frac {\sqrt {2-b\,x}\,\left (\frac {b\,x}{3}+\frac {1}{3}\right )}{x^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(5/2)*(2 - b*x)^(1/2)),x)

[Out]

-((2 - b*x)^(1/2)*((b*x)/3 + 1/3))/x^(3/2)

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